6) Find the Greatest Common Divisor of 35, 60, and 75 by the List Method, Factor Tree Division, and Prime Factors Method. 7) Find the Least Common Multiple of 60 and 75 using either the Dividing by Prime Factors Method or the Formula.
The Correct Answer and Explanation is:
6) Finding the Greatest Common Divisor (GCD) of 35, 60, and 75
Method 1: List Method
To find the GCD, we first list the factors of each number.
- Factors of 35: 1, 5, 7, 35
- Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- Factors of 75: 1, 3, 5, 15, 25, 75
Now, identify the common factors:
- The common factors of 35, 60, and 75 are 1 and 5.
The greatest common factor is 5.
Method 2: Factor Tree Division
We begin by breaking each number down into its prime factors.
- 35: 35 → 5 × 7
- 60: 60 → 2 × 2 × 3 × 5
- 75: 75 → 3 × 5 × 5
The common prime factor across all three numbers is 5.
Therefore, the GCD is 5.
Method 3: Prime Factors Method
From the prime factorization:
- 35 = 5 × 7
- 60 = 2² × 3 × 5
- 75 = 3 × 5²
The only common prime factor is 5, and the lowest power of 5 is 5¹.
Thus, the GCD is 5.
7) Finding the Least Common Multiple (LCM) of 60 and 75
Method 1: Dividing by Prime Factors Method
First, find the prime factorizations of the numbers:
- 60 = 2² × 3 × 5
- 75 = 3 × 5²
To find the LCM, take the highest powers of all the prime factors:
- Highest power of 2: 2²
- Highest power of 3: 3¹
- Highest power of 5: 5²
Now, multiply these together:LCM=22×3×52=4×3×25=300LCM = 2² × 3 × 5² = 4 × 3 × 25 = 300LCM=22×3×52=4×3×25=300
So, the LCM of 60 and 75 is 300.
Method 2: Formula Method
The formula for the LCM using the GCD is:LCM(a,b)=∣a×b∣GCD(a,b)LCM(a, b) = \frac{|a × b|}{GCD(a, b)}LCM(a,b)=GCD(a,b)∣a×b∣
Using the previously calculated GCD of 60 and 75 (which is 15):LCM(60,75)=∣60×75∣15=450015=300LCM(60, 75) = \frac{|60 × 75|}{15} = \frac{4500}{15} = 300LCM(60,75)=15∣60×75∣=154500=300
Thus, the LCM is 300.
Explanation:
- GCD is the largest number that divides both numbers without leaving a remainder. It shows the greatest shared factor between them.
- LCM is the smallest number that both numbers can divide without leaving a remainder. It’s useful for finding common multiples, especially when working with fractions, ratios, or scheduling problems.
